Page:LorentzGravitation1916.djvu/28

direction of one of the coordinates e. g. of xe{\displaystyle x_{e}} over the distance dxe{\displaystyle dx_{e}}. We had then to keep in mind that for the two sides the values of ub{\displaystyle u_{b}}, which have opposite signs, are a little different; and it was precisely this difference that was of importance. In the calculation of the integral

however it may be neglected. Hence, when we express the components ub{\displaystyle u_{b}} in terms of the quantities ψab{\displaystyle \psi _{ab}}, we may give to these latter the values which they have at the point P{\displaystyle P}.

Let us consider two sides situated at the ends of the edges dxe{\displaystyle dx_{e}} and whose magnitude we may therefore express in x{\displaystyle x}-units dxjdxkdxl{\displaystyle dx_{j}dx_{k}dx_{l}} if j,k,l{\displaystyle j,k,l} are the numbers which are left of 1, 2, 3, 4 when the number e{\displaystyle e} is omitted. For the part contributed to (38) by the side Σ2{\displaystyle \Sigma _{2}} we found in § 26

where the first integral relates to Σ2{\displaystyle \Sigma _{2}} and the second to Σ1{\displaystyle \Sigma _{1}}. It is clear that but one value of c{\displaystyle c}, viz. e{\displaystyle e} has to be considered. As everywhere in Σ1:xc=0{\displaystyle \Sigma _{1}:\mathrm {x} _{c}=0} and everywhere in Σ2:xc=dxe{\displaystyle \Sigma _{2}:\mathrm {x} _{c}=dx_{e}} it is further evident that the above expression becomes

This is one part contributed to the expression (36). A second part, the origin of which will be immediately understood, is found by interchanging b{\displaystyle b} and e{\displaystyle e}. With a view to (37) and because of

ψeb=−ψbe{\displaystyle \psi {}_{eb}=-\psi {}_{be}}

we have for each term of (36) another by which it is cancelled. This is what had to be proved.

§ 31. Now that we have shown that equation (32) holds for each element (dx1,…dx4){\displaystyle \left(dx_{1},\dots dx_{4}\right)} we may conclude by the considerations of § 21 that this is equally true for any arbitrarily chosen magnitude and shape of the extension Ω{\displaystyle \Omega }. In particular the equation may be applied to an element (dx1′,…dx4′){\displaystyle \left(dx'_{1},\dots dx'_{4}\right)} and by considerations exactly similar to